배우다 = to learn (a skill, subject, way to do something, etc).
학습하다 = similar to 배우다, but more formal and it fits academic contexts better. (학습 = n. learning)
공부하다 = to study (a subject, as a student or to acquire practical knowledge). (공부 = study)
배우다 and 학습하다 are typically used with an object. 공부하다 can be used with or without one.
배우다 and 공부하다 are widely used everyday word. 공부 is very common both as a verb and noun.
학습하다 is used less frequently.
학습 by itself is common as a noun and as part of compound nouns or phrases (학습 시간, 학습 효과, 학습 목표, 학습 방법, 학습량, 학습 연령, ...).
- 친구는 영어를 배우러 학원에 다닌다 = My friend goes to a private school to learn English.
- 초등학생의 학습량이 너무 많다고 한다 = They say grade-schoolers have too much to study.
- 인생의 모든 것이 공부로 결정되지는 않는다 = Not everything in life is determined by what you study.